Interrelations among problems

As it is obvious in Figure 6.2, to do routing problem, feasible range of TCT must be defined first, and this feasible range is part of scheduling problem. Moreover, to define early start which is part of scheduling problem, loading problem must be considered. However, loading problem

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cannot be done without knowing the TCTs found in scheduling problem. Furthermore, the optimal routing needs information about the values of the objective function from loading problem (inventory costs and CV), and also the minimum possible number of trains which must be found based on the feasible range found in scheduling problem. All these relations reveal the importance of investigating the three problems together in parallel.

6.8 Results and analysis

In the example presented, at first, the weight of the objective function in equation (6.21) was set to be 100 for the CV value since it is usually very small and this value should affect the results. Each one in the other two terms in the objective function was given a weight of 1. The time buffer among train cycles was set to be 1 station cycle. All the problems were run together and needed just few seconds on a normal personal computer. The results of the example are in table 6.3 where the feasible space is in the left-hand side of the table. The value of 1 is for the active cells in the feasible space. For example it is shown in table 6.3 that the final solution must have a cell containing the first 5, 6 or 7 stations. In the optimal solution on the right side of the table, there are three trains, and one of them supplies the stations from 1 to 7, the second one supplies the stations from 8 to 14, and the last one supplies the rest of the stations. The scheduling results are shown in the right side of table 6.3 where the results are expressed in the form (x, y) where x is the TCT and y is the early start. There was no early start or early loading at all in the optimal solution. However, early loading occurs in the third cell if the weight of CV value is increased to be 102, and the results will contain three cells, where the first one contains the stations from 1 to 6, the second cell contains the stations from 7 to 13, and the third cell contains the remaining stations. These changes in results show the effect of the objective of decreasing the variability in loading on the results of the three problems of routing, scheduling, and loading. The effect on routing is not in the number of trains but in the cells formation (the assignment of stations to cells).

Table 6.4 shows the results when the time buffer is set to be zero where the value of the objective function is decreased from 487.3 to 452.1 and the feasible space is increased to contain more cells. Furthermore, the last cell from station 13 to station 20 has early loading and also early start of 1 station cycle. The first route is too short compared to the other two ones. To fix this problem, the weight w1 in equation (6.21) must be increased. If it is set to be 21 for example, the

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new solution will contain a cell from station 1 to 6, another cell contains stations from 7 to 12, and the last one contains the other stations. However, if this weight is increased to be 67, the first cell will be from station 1 to 7, the second cell will be from station 8 to 13, and the last cell will contain the remaining stations.

Table 6.3 Feasible space and optimal solution when time buffer is 1 station cycle

Feasible space Optimal solution

Stations Station cycle Stations Station cycle 5 6 7 13 14 20 7 14 20 1 1 1 1 1 (10, 0)* 6 1 6 7 1 7 8 1 1 8 (10, 0) 14 1 14 15 1 15 (9, 0) *

(x, y): x is the TCT and y is the early start

Table 6.4 Feasible space and optimal solution when time buffer is 0

Feasible space Optimal solution

Stations Station cycle Stations Station cycle 4 5 6 7 8 12 13 14 15 16 20 4 12 20 1 1 1 1 1 1 1 (6, 0) 5 1 5 (10, 0) 6 1 1 6 7 1 1 7 8 1 1 1 8 9 1 1 1 1 1 9 13 1 13 (10, -1) 14 1 14 15 1 15 16 1 16 17 1 17

Table 6.5 shows several trials in which the K and KLS values were changed to get the optimal objective function value and the number of needed trains. The 4th trial represents the current status. If the K value is decreased to be 10, four trains are needed. It is noted that no matter how much the K value is increased more than 16, it cannot affect the results. This is because of the

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fixed value of KLS. So to get lower number of trains, the KLS value must be increased as in the 10th trial. This result is important since it shows that it is not very helpful to get tugger trains with high capacities if the KLS cannot be increased.

Table 6.5 Effect of K and KLS values on the results

Trial K KLS Number of trains O.F. value

1 17 3 3 474.96 2 16 3 3 474.96 3 15 3 3 452.12 4 14 3 3 452.12 5 13 3 3 526.17 6 12 3 3 592.30 7 11 3 3 600.93 8 10 3 4 548.33 9 14 4 3 452.12 10 21 4 2 630.67 6.9 Summary

In this chapter, the routing, scheduling, and loading problems of the tow train were investigated together in parallel in decentralized supermarket system. The objective function decreases the number of trains, variability in loading and in route lengths, and inventory holding costs in normal and in early loading. This was done using analytical equations, DP, and IP techniques. Constraints related to tugger capacity, capacity of line-side area, routes times, and time buffer among train cycles were taken into consideration. Besides the time buffer, safety stock and empty space capacity for line-side inventory can be used. Further investigation about loading problem was presented using IP.

This chapter shows the importance of studying the three problems together since they are interrelated, and this parallel investigation should be done to minimize the total inventory holding costs to the minimum possible value and to take into account decreasing the variability in the loaded quantities. It also shows the effect of time buffer on the feasible solution space. Moreover, it shows the importance of considering the capacity of line-side area and the capacity of the train in the same time, where getting tugger trains with so high capacities does not enhance the system because the line-side area is usually limited to small amounts. It also shows the

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importance of the objective regarding decreasing the variability in loaded quantities since it affects all the results of routing, scheduling, and loading.

This chapter is a part of investigating the demand-oriented system in which the demand of stations for bins is known at least for the next shift. In the next chapter, further investigation is done for this demand-oriented system in terms of scheduling, and loading problems to accommodate the capacity problems of trains and line-side area. In chapter 8, routing, scheduling, and loading will be investigated one more time to accommodate assembly line disturbances. In chapter 7 and 8, the environment investigated contains both the centralized and decentralized supermarket system.

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